Integral Separation, Lyapunov Transformations, and the Small Change in the Direction of the Solutions of Periodic Linear Systems of Ordinary Differential Equations

Abstract

First, we discuss the relation between the integral separation of solutions of a linear periodic systemx′=A(t)x(1)and the structure of its characteristic multipliers. Second, we investigate the reducibility of (1) by a complex or a real Lyapunov transformation. Third, we prove the property of the small change in the direction of solutions of ω-periodic system (1) as the result of a small perturbation of (1) under the following assumptions:(a)When the characteristic multipliers of (1) are distinct and the perturbation of (1) is admissible. The perturbationC(t)=B(t)−A(t) is calledadmissibleto the system (1), if the characteristic multipliers ρkand ρkof the systems (1) andy′=B(t)y,(2)respectively, have the same argument, i.e., argρk=argρk,k=1,…,n. In this case, we also say that the perturbed ω-periodic system (2) is admissible to the system (1).(b)When the characteristic multipliers of (1) are real and distinct and the perturbation is arbitrary.